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OCR M4 2012 June Q1
6 marks Standard +0.8
1 A uniform square lamina, of mass 4.5 kg and side 0.6 m , is rotating about a fixed vertical axis which is perpendicular to the lamina and passes through its centre. A stationary particle becomes attached to the lamina at one of its corners, and this causes the angular speed of the lamina to change instantaneously from \(2.2 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to \(1.5 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
  1. Find the mass of the particle. The lamina then slows down with constant angular deceleration. It turns through 36 radians as its angular speed reduces from \(1.5 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to zero.
  2. Find the time taken for the lamina to come to rest.
OCR M4 2012 June Q2
7 marks Challenging +1.2
2 A uniform solid of revolution is formed by rotating the region bounded by the \(x\)-axis and the curve \(y = x \left( 1 - \frac { x ^ { 2 } } { a ^ { 2 } } \right)\) for \(0 \leqslant x \leqslant a\), where \(a\) is a constant, about the \(x\)-axis. Find the \(x\)-coordinate of the centre of mass of this solid.
OCR M4 2012 June Q3
10 marks Challenging +1.2
3 \includegraphics[max width=\textwidth, alt={}, center]{ab760a4b-e0ec-4256-838f-ed6c762ff18b-2_460_388_1160_826} A ship \(S\) is travelling with constant velocity \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a course with bearing \(120 ^ { \circ }\). A patrol boat \(B\) observes the ship when \(S\) is due north of \(B\). The patrol boat \(B\) then moves with constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line (see diagram).
  1. Given that \(V = 18\), find the bearing of the course of \(B\) such that \(B\) intercepts \(S\).
  2. Given instead that \(V = 9\), find the bearing of the course of \(B\) such that \(B\) passes as close as possible to \(S\).
  3. Find the smallest value of \(V\) for which it is possible for \(B\) to intercept \(S\).
OCR M4 2012 June Q4
7 marks Challenging +1.2
4 A uniform lamina of mass 18 kg occupies the region bounded by the \(x\)-axis, the \(y\)-axis, the line \(x = \ln 9\) and the curve \(y = \mathrm { e } ^ { \frac { 1 } { 2 } x }\) for \(0 \leqslant x \leqslant \ln 9\). The unit of length is the metre. Find the moment of inertia of this lamina about the \(x\)-axis.
OCR M4 2012 June Q5
15 marks Challenging +1.3
5 A uniform rod of mass 4 kg and length 2.4 m can rotate in a vertical plane about a fixed horizontal axis through one end of the rod. The rod is released from rest in a horizontal position and a frictional couple of constant moment 20 Nm opposes the motion.
  1. Find the angular acceleration of the rod immediately after it is released.
  2. Find the angle that the rod makes with the horizontal when its angular acceleration is zero.
  3. Find the maximum angular speed of the rod.
  4. The rod first comes to instantaneous rest after rotating through an angle \(\theta\) radians from its initial position. Find an equation for \(\theta\), and verify that \(2.0 < \theta < 2.1\).
OCR M4 2012 June Q6
12 marks Challenging +1.8
6 \includegraphics[max width=\textwidth, alt={}, center]{ab760a4b-e0ec-4256-838f-ed6c762ff18b-3_716_483_890_790} Two small smooth pegs \(P\) and \(Q\) are fixed at a distance \(2 a\) apart on the same horizontal level, and \(A\) is the mid-point of \(P Q\). A light rod \(A B\) of length \(4 a\) is freely pivoted at \(A\) and can rotate in the vertical plane containing \(P Q\), with \(B\) below the level of \(P Q\). A particle of mass \(m\) is attached to the rod at \(B\). A light elastic string, of natural length \(2 a\) and modulus of elasticity \(\lambda\), passes round the pegs \(P\) and \(Q\) and its two ends are attached to the rod at the point \(X\), where \(A X = a\). The angle between the rod and the downward vertical is \(\theta\), where \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\) (see diagram). You are given that the elastic energy stored in the string is \(\lambda a ( 1 + \cos \theta )\).
  1. Show that \(\theta = 0\) is a position of equilibrium, and show that the equilibrium is stable if \(\lambda < 4 m g\).
  2. Given that \(\lambda = 3 m g\), show that \(\ddot { \theta } = - k \frac { g } { a } \sin \theta\), stating the value of the constant \(k\). Hence find the approximate period of small oscillations of the system about the equilibrium position \(\theta = 0\).
OCR M4 2012 June Q7
15 marks Challenging +1.2
7 \includegraphics[max width=\textwidth, alt={}, center]{ab760a4b-e0ec-4256-838f-ed6c762ff18b-4_783_783_255_641} A uniform circular disc with centre \(C\) has mass \(m\) and radius \(a\). The disc is free to rotate in a vertical plane about a fixed horizontal axis passing through a point \(A\) on the disc, where \(A C = \frac { 1 } { 2 } a\). The disc is slightly disturbed from rest in the position with \(C\) vertically above \(A\). When \(A C\) makes an angle \(\theta\) with the upward vertical the force exerted by the axis on the disc has components \(R\) parallel to \(A C\) and \(S\) perpendicular to \(A C\) (see diagram).
  1. Show that the angular speed of the disc is \(\sqrt { \frac { 4 g ( 1 - \cos \theta ) } { 3 a } }\).
  2. Find the angular acceleration of the disc, in terms of \(a , g\) and \(\theta\).
  3. Find \(R\) and \(S\), in terms of \(m , g\) and \(\theta\).
  4. Find the magnitude of the force exerted by the axis on the disc at an instant when \(R = 0\).
OCR M4 2013 June Q1
5 marks Moderate -0.8
1 A camshaft inside an engine is rotating with angular speed \(42 \mathrm { rads } ^ { - 1 }\). When the throttle is opened the camshaft speeds up with constant angular acceleration, and 8 seconds after the throttle was opened the angular speed is \(76 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
  1. Find the angular acceleration of the camshaft.
  2. Find the time taken for the camshaft to turn through 810 radians from the moment that the throttle was opened.
OCR M4 2013 June Q2
7 marks Standard +0.8
2 A straight \(\operatorname { rod } A B\) has length \(a\). The rod has variable density, and at a distance \(x\) from \(A\) its mass per unit length is given by \(k \left( 4 - \sqrt { \frac { x } { a } } \right)\), where \(k\) is a constant. Find the distance from \(A\) of the centre of mass of the rod.
OCR M4 2013 June Q3
8 marks Challenging +1.2
3 The region \(R\) is bounded by the \(x\)-axis, the \(y\)-axis, the curve \(y = a \mathrm { e } ^ { \frac { x } { a } }\) and the line \(x = a \ln 2\) (where \(a\) is a positive constant). A uniform solid of revolution, of mass \(M\), is formed by rotating \(R\) through \(2 \pi\) radians about the \(x\)-axis. Find, in terms of \(M\) and \(a\), the moment of inertia about the \(x\)-axis of this solid of revolution.
[0pt] [8]
OCR M4 2013 June Q4
12 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{6e3d5f5e-7ffa-4111-903d-468fb4d20192-2_364_1313_1224_376} An unidentified aircraft \(U\) is flying horizontally with constant velocity \(250 \mathrm {~ms} ^ { - 1 }\) in the direction with bearing \(040 ^ { \circ }\). Two spotter planes \(P\) and \(Q\) are flying horizontally at the same height as \(U\), and at one instant \(P\) is 15000 m due west of \(U\), and \(Q\) is 15000 m due east of \(U\) (see diagram).
  1. Plane \(P\) is flying with constant velocity \(210 \mathrm {~ms} ^ { - 1 }\) in the direction with bearing \(070 ^ { \circ }\).
OCR M4 2013 June Q6
12 marks Challenging +1.2
6 \includegraphics[max width=\textwidth, alt={}, center]{6e3d5f5e-7ffa-4111-903d-468fb4d20192-4_640_608_267_715} A smooth wire forms a circle with centre \(O\) and radius \(a\), and is fixed in a vertical plane. The highest point on the wire is \(A\). A small ring \(R\) of mass \(m\) moves along the wire. A light elastic string, with natural length \(\frac { 1 } { 2 } a\) and modulus of elasticity \(2 m g\), has one end attached to \(A\) and the other end attached to \(R\). The string \(A R\) makes an angle \(\theta\) (measured anticlockwise) with the downward vertical (see diagram), and you may assume that the string does not become slack.
  1. Taking \(A\) as the reference level for gravitational potential energy, show that the total potential energy of the system is \(m g a \left( 6 \cos ^ { 2 } \theta - 4 \cos \theta + \frac { 1 } { 2 } \right)\).
  2. Show that there are two positions of equilibrium for which \(0 \leqslant \theta < \frac { 1 } { 2 } \pi\).
  3. For each of these positions of equilibrium, determine whether it is stable or unstable.
OCR M4 2013 June Q7
14 marks Challenging +1.8
7 \includegraphics[max width=\textwidth, alt={}, center]{6e3d5f5e-7ffa-4111-903d-468fb4d20192-5_584_686_264_678} \(A B C D\) is a uniform rectangular lamina with mass \(m\) and sides \(A B = 6 a\) and \(A D = 8 a\). The lamina rotates freely in a vertical plane about a fixed horizontal axis passing through \(A\), and it is released from rest in the position with \(D\) vertically above \(A\). When the diagonal \(A C\) makes an angle \(\theta\) below the horizontal, the force acting on the lamina at \(A\) has components \(R\) parallel to \(C A\) and \(S\) perpendicular to \(C A\) (see diagram).
  1. Find the moment of inertia of the lamina about the axis through \(A\), in terms of \(m\) and \(a\).
  2. Show that the angular speed of the lamina is \(\sqrt { \frac { 3 g ( 4 + 5 \sin \theta ) } { 50 a } }\).
  3. Find the angular acceleration of the lamina, in terms of \(a , g\) and \(\theta\).
  4. Find \(R\) and \(S\), in terms of \(m , g\) and \(\theta\).
OCR M4 2014 June Q1
7 marks Challenging +1.2
1 Alan is running in a straight line on a bearing of \(090 ^ { \circ }\) at a constant speed of \(4 \mathrm {~ms} ^ { - 1 }\). Ben sees Alan when they are 50 m apart and Alan is on a bearing of \(060 ^ { \circ }\) from Ben. Ben sets off immediately to intercept Alan by running at a constant speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Calculate the bearing on which Ben should run to intercept Alan.
  2. Calculate the magnitude of the velocity of Ben relative to Alan and find the time it takes, from the moment Ben sees Alan, for Ben to intercept Alan.
OCR M4 2014 June Q2
11 marks Challenging +1.2
2 A uniform solid circular cone has mass \(M\) and base radius \(R\).
  1. Show by integration that the moment of inertia of the cone about its axis of symmetry is \(\frac { 3 } { 10 } M R ^ { 2 }\). (You may assume the standard formula \(\frac { 1 } { 2 } m r ^ { 2 }\) for the moment of inertia of a uniform disc about its axis and that the volume of a cone is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).) The axis of symmetry of the cone is fixed vertically and the cone is rotating about its axis at an angular speed of \(6 \mathrm { rad } \mathrm { s } ^ { - 1 }\). A frictional couple of constant moment 0.027 Nm is applied to the cone bringing it to rest. Given that the mass of the cone is 2 kg and its base radius is 0.3 m , find
  2. the constant angular deceleration of the cone,
  3. the time taken for the cone to come to rest from the instant that the couple is applied.
OCR M4 2014 June Q3
8 marks Challenging +1.2
3 The region bounded by the \(y\)-axis and the curves \(y = \sin 2 x\) and \(y = \sqrt { 2 } \cos x\) for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\) is occupied by a uniform lamina. Find the exact value of the \(x\)-coordinate of the centre of mass of the lamina.
OCR M4 2014 June Q4
13 marks Challenging +1.8
4 A uniform square lamina has mass \(m\) and sides of length \(2 a\).
  1. Calculate the moment of inertia of the lamina about an axis through one of its corners perpendicular to its plane. \includegraphics[max width=\textwidth, alt={}, center]{639c658e-0aca-4161-9e77-0f4c494b0b55-3_693_640_434_715} The uniform square lamina has centre \(C\) and is free to rotate in a vertical plane about a fixed horizontal axis passing through one of its corners \(A\). The lamina is initially held such that \(A C\) is vertical with \(C\) above \(A\). The lamina is slightly disturbed from rest from this initial position. When \(A C\) makes an angle \(\theta\) with the upward vertical, the force exerted by the axis on the lamina has components \(X\) parallel to \(A C\) and \(Y\) perpendicular to \(A C\) (see diagram).
  2. Show that the angular speed, \(\omega\), of the lamina satisfies \(a \omega ^ { 2 } = \frac { 3 } { 4 } g \sqrt { 2 } ( 1 - \cos \theta )\).
  3. Find \(X\) and \(Y\) in terms of \(m , g\) and \(\theta\). \section*{Question 5 begins on page 4.}
    \includegraphics[max width=\textwidth, alt={}]{639c658e-0aca-4161-9e77-0f4c494b0b55-4_767_337_248_863}
    A pendulum consists of a uniform rod \(A B\) of length \(4 a\) and mass \(4 m\) and a spherical shell of radius \(a\), mass \(m\) and centre \(C\). The end \(B\) of the rod is rigidly attached to a point on the surface of the shell in such a way that \(A B C\) is a straight line. The pendulum is initially at rest with \(B\) vertically below \(A\) and it is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through \(A\) (see diagram).
  1. Show that the moment of inertia of the pendulum about the axis of rotation is \(47 m a ^ { 2 }\). A particle of mass \(m\) is moving horizontally in the plane in which the pendulum is free to rotate. The particle has speed \(\sqrt { k g a }\), where \(k\) is a positive constant, and strikes the rod at a distance \(3 a\) from \(A\). In the subsequent motion the particle adheres to the rod and the combined rigid body \(P\) starts to rotate.
  2. Show that the initial angular speed of \(P\) is \(\frac { 3 } { 56 } \sqrt { \frac { k g } { a } }\).
  3. For the case \(k = 4\), find the angle that \(P\) has turned through when \(P\) first comes to instantaneous rest.
  4. Find the least value of \(k\) such that the rod reaches the horizontal. \includegraphics[max width=\textwidth, alt={}, center]{639c658e-0aca-4161-9e77-0f4c494b0b55-5_437_903_269_573} A uniform rod \(A B\) has mass \(m\) and length \(2 a\). The rod can rotate in a vertical plane about a smooth fixed horizontal axis passing through \(A\). One end of a light elastic string of natural length \(a\) and modulus of elasticity \(\sqrt { 3 } m g\) is attached to \(A\). The string passes over a small smooth fixed pulley \(C\), where \(A C\) is horizontal and \(A C = a\). The other end of the string is attached to the rod at its mid-point \(D\). The rod makes an angle \(\theta\) below the horizontal (see diagram).
  1. Taking \(A\) as the reference level for gravitational potential energy, show that the total potential energy \(V\) of the system is given by $$V = m g a ( \sqrt { 3 } - \sin \theta - \sqrt { 3 } \cos \theta ) .$$
  2. Show that \(\theta = \frac { 1 } { 6 } \pi\) is a position of stable equilibrium for the system. The system is making small oscillations about the equilibrium position.
  3. By differentiating the energy equation with respect to time, show that $$\frac { 4 } { 3 } a \ddot { \theta } = g ( \cos \theta - \sqrt { 3 } \sin \theta ) .$$
  4. Using the substitution \(\theta = \phi + \frac { 1 } { 6 } \pi\), show that the motion is approximately simple harmonic, and find the approximate period of the oscillations. \section*{END OF QUESTION PAPER}
OCR M4 2015 June Q1
5 marks Moderate -0.8
1 A turntable is rotating at \(3 \mathrm { rad } \mathrm { s } ^ { - 1 }\). The turntable is then accelerated so that after 4 revolutions it is rotating at \(12.4 \mathrm { rad } \mathrm { s } ^ { - 1 }\). Assuming that the angular acceleration of the turntable is constant,
  1. find the angular acceleration,
  2. find the time taken to increase its angular speed from \(3 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to \(12.4 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
OCR M4 2015 June Q2
10 marks Standard +0.8
2 The region bounded by the \(x\)-axis, the lines \(x = 1\) and \(x = 2\), and the curve \(y = k x ^ { 2 }\), where \(k\) is a positive constant, is occupied by a uniform lamina.
  1. Find the exact \(x\)-coordinate of the centre of mass of the lamina.
  2. Given that the \(x\) - and \(y\)-coordinates of the centre of mass of the lamina are equal, find the exact value of \(k\).
OCR M4 2015 June Q3
11 marks Standard +0.8
3 Two planes, \(A\) and \(B\), flying at the same altitude, are participating in an air show. Initially the planes are 400 m apart and plane \(B\) is on a bearing of \(130 ^ { \circ }\) from plane \(A\). Plane \(A\) is moving due south with a constant speed of \(75 \mathrm {~ms} ^ { - 1 }\). Plane \(B\) is moving at a constant speed of \(40 \mathrm {~ms} ^ { - 1 }\) and has set a course to get as close as possible to \(A\).
  1. Find the bearing of the course set by \(B\) and the shortest distance between the two planes in the subsequent motion.
  2. Find the total distance travelled by \(A\) and \(B\) from the instant when they are initially 400 m apart to the point of their closest approach.
OCR M4 2015 June Q4
9 marks Challenging +1.8
4
  1. Write down the moment of inertia of a uniform circular disc of mass \(m\) and radius \(2 a\) about a diameter. A uniform solid cylinder has mass \(M\), radius \(2 r\) and height \(h\).
  2. Show by integration, and using the result from part (i), that the moment of inertia of the cylinder about a diameter of an end face is $$M \left( r ^ { 2 } + \frac { 1 } { 3 } h ^ { 2 } \right)$$ and hence find the moment of inertia of the cylinder about a diameter through the centre of the cylinder. \includegraphics[max width=\textwidth, alt={}, center]{4b50b084-081f-48d2-ad5b-95b2c9e55dfc-3_919_897_260_591} A smooth circular wire hoop, with centre \(O\) and radius \(r\), is fixed in a vertical plane. The highest point on the wire is \(H\). A small bead \(B\) of mass \(m\) is free to move along the wire. A light inextensible string of length \(a\), where \(a > 2 r\), has one end attached to the bead. The other end of the string passes over a small smooth pulley at \(H\) and carries at its end a particle \(P\) of mass \(\lambda m\), where \(\lambda\) is a positive constant. The part of the string \(H P\) is vertical and the part of the string \(B H\) makes an angle \(\theta\) radians with the downward vertical where \(0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi\) (see diagram). You may assume that \(P\) remains above the lowest point of the wire.
OCR M4 2015 June Q6
22 marks Challenging +1.8
6 A pendulum consists of a uniform rod \(A B\) of length \(2 a\) and mass \(2 m\) and a particle of mass \(m\) that is attached to the end \(B\). The pendulum can rotate in a vertical plane about a smooth fixed horizontal axis passing through \(A\).
  1. Show that the moment of inertia of this pendulum about the axis of rotation is \(\frac { 20 } { 3 } m a ^ { 2 }\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{4b50b084-081f-48d2-ad5b-95b2c9e55dfc-4_572_86_852_575} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{4b50b084-081f-48d2-ad5b-95b2c9e55dfc-4_582_456_842_1050} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The pendulum is initially held with \(B\) vertically above \(A\) (see Fig.1) and it is slightly disturbed from this position. When the angle between the pendulum and the upward vertical is \(\theta\) radians the pendulum has angular speed \(\omega \mathrm { rads } ^ { - 1 }\) (see Fig. 2).
  2. Show that $$\omega ^ { 2 } = \frac { 6 g } { 5 a } ( 1 - \cos \theta ) .$$
  3. Find the angular acceleration of the pendulum in terms of \(g , a\) and \(\theta\). At an instant when \(\theta = \frac { 1 } { 3 } \pi\), the force acting on the pendulum at \(A\) has magnitude \(F\).
  4. Find \(F\) in terms of \(m\) and \(g\). It is given that \(a = 0.735 \mathrm {~m}\).
  5. Show that the time taken for the pendulum to move from the position \(\theta = \frac { 1 } { 6 } \pi\) to the position \(\theta = \frac { 1 } { 3 } \pi\) is given by $$k \int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \operatorname { cosec } \left( \frac { 1 } { 2 } \theta \right) \mathrm { d } \theta ,$$ stating the value of the constant \(k\). Hence find the time taken for the pendulum to rotate between these two points. (You may quote an appropriate result given in the List of Formulae (MF1).) \section*{END OF QUESTION PAPER}
OCR M4 2017 June Q1
7 marks Challenging +1.2
1 A uniform rod with centre \(C\) has mass \(2 M\) and length 4a. The rod is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through a point \(A\) on the rod, where \(A C = k a\) and \(0 < k < 2\). The rod is making small oscillations about the equilibrium position with period \(T\).
  1. Show that \(T = 2 \pi \sqrt { \frac { a } { 3 g } \left( \frac { 4 + 3 k ^ { 2 } } { k } \right) }\). (You may assume the standard formula \(T = 2 \pi \sqrt { \frac { I } { m g h } }\) for the period of small oscillations of a compound pendulum.)
  2. Hence find the value of \(k ^ { 2 }\) for which the period of oscillations is least.
OCR M4 2017 June Q2
9 marks Challenging +1.2
2 A ship \(S\) is travelling with constant speed \(5 \mathrm {~ms} ^ { - 1 }\) on a course with bearing \(325 ^ { \circ }\). A second ship \(T\) observes \(S\) when \(S\) is 9500 m from \(T\) on a bearing of \(060 ^ { \circ }\) from \(T\). Ship \(T\) sets off in pursuit, travelling with constant speed \(8.5 \mathrm {~ms} ^ { - 1 }\) in a straight line.
  1. Find the bearing of the course which \(T\) should take in order to intercept \(S\).
  2. Find the distance travelled by \(S\) from the moment that \(T\) sets off in pursuit until the point of interception.
OCR M4 2017 June Q3
17 marks Challenging +1.2
3 \includegraphics[max width=\textwidth, alt={}, center]{57323af2-8cf3-4721-b2c8-a968264be343-2_439_444_1318_822} A uniform rod \(A B\) has mass \(m\) and length \(4 a\). The rod can rotate in a vertical plane about a smooth fixed horizontal axis passing through \(A\). One end of a light elastic string of natural length \(a\) and modulus of elasticity \(\lambda m g\) is attached to \(B\). The other end of the string is attached to a small light ring which slides on a fixed smooth horizontal rail which is in the same vertical plane as the rod. The rail is a vertical distance \(3 a\) above \(A\). The string is always vertical and the rod makes an angle \(\theta\) radians with the horizontal, where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\) (see diagram).
  1. Taking \(A\) as the reference level for gravitational potential energy, find an expression for the total potential energy \(V\) of the system, and show that $$\frac { \mathrm { d } V } { \mathrm {~d} \theta } = 2 m g a \cos \theta ( 4 \lambda ( 1 + 2 \sin \theta ) - 1 ) .$$ Determine the positions of equilibrium and the nature of their stability in the cases
  2. \(\lambda > \frac { 1 } { 12 }\),
  3. \(\lambda < \frac { 1 } { 12 }\). \includegraphics[max width=\textwidth, alt={}, center]{57323af2-8cf3-4721-b2c8-a968264be343-3_392_689_269_671} The diagram shows the curve with equation \(y = \frac { 1 } { 2 } \ln x\). The region \(R\), shaded in the diagram, is bounded by the curve, the \(x\)-axis and the line \(x = 4\). A uniform solid of revolution is formed by rotating \(R\) completely about the \(y\)-axis to form a solid of volume \(V\).
  1. Show that \(V = \frac { 1 } { 4 } \pi ( 64 \ln 2 - 15 )\).
  2. Find the exact \(y\)-coordinate of the centre of mass of the solid. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{57323af2-8cf3-4721-b2c8-a968264be343-4_385_741_269_646} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} Fig. 1 shows part of the line \(y = \frac { a } { h } x\), where \(a\) and \(h\) are constants. The shaded region bounded by the line, the \(x\)-axis and the line \(x = h\) is rotated about the \(x\)-axis to form a uniform solid cone of base radius \(a\), height \(h\) and volume \(\frac { 1 } { 3 } \pi a ^ { 2 } h\). The mass of the cone is \(M\).
  1. Show by integration that the moment of inertia of the cone about the \(y\)-axis is \(\frac { 3 } { 20 } M \left( a ^ { 2 } + 4 h ^ { 2 } \right)\). (You may assume the standard formula \(\frac { 1 } { 4 } m r ^ { 2 }\) for the moment of inertia of a uniform disc about a diameter.) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{57323af2-8cf3-4721-b2c8-a968264be343-4_501_556_1238_726} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} A uniform solid cone has mass 3 kg , base radius 0.4 m and height 1.2 m . The cone can rotate about a fixed vertical axis passing through its centre of mass with the axis of the cone moving in a horizontal plane. The cone is rotating about this vertical axis at an angular speed of \(9.6 \mathrm { rad } \mathrm { s } ^ { - 1 }\). A stationary particle of mass \(m \mathrm {~kg}\) becomes attached to the vertex of the cone (see Fig. 2). The particle being attached to the cone causes the angular speed to change instantaneously from \(9.6 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to \(7.8 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
  2. Find the value of \(m\). \includegraphics[max width=\textwidth, alt={}, center]{57323af2-8cf3-4721-b2c8-a968264be343-5_534_501_255_767} A triangular frame \(A B C\) consists of three uniform rods \(A B , B C\) and \(C A\), rigidly joined at \(A , B\) and \(C\). Each rod has mass \(m\) and length \(2 a\). The frame is free to rotate in a vertical plane about a fixed horizontal axis passing through \(A\). The frame is initially held such that the axis of symmetry through \(A\) is vertical and \(B C\) is below the level of \(A\). The frame starts to rotate with an initial angular speed of \(\omega\) and at time \(t\) the angle between the axis of symmetry through \(A\) and the vertical is \(\theta\) (see diagram).
  1. Show that the moment of inertia of the frame about the axis through \(A\) is \(6 m a ^ { 2 }\).
  2. Show that the angular speed \(\dot { \theta }\) of the frame when it has turned through an angle \(\theta\) satisfies $$a \dot { \theta } ^ { 2 } = a \omega ^ { 2 } - k g \sqrt { 3 } ( 1 - \cos \theta ) ,$$ stating the exact value of the constant \(k\).
    Hence find, in terms of \(a\) and \(g\), the set of values of \(\omega ^ { 2 }\) for which the frame makes complete revolutions. At an instant when \(\theta = \frac { 1 } { 6 } \pi\), the force acting on the frame at \(A\) has magnitude \(F\).
  3. Given that \(\omega ^ { 2 } = \frac { 2 g } { a \sqrt { 3 } }\), find \(F\) in terms of \(m\) and \(g\). \section*{END OF QUESTION PAPER}